Progressive Meshes with Controlled Topology Modification

University of Bonn

Institute II. for Computer Science

Computer Graphics Group

Pavcl Borodin Rchinhard Klcin

Introduction

● Progressive Meshes

Which was introduced by Hoppe(1996)

● Retain the topology of original model

Vertex Contraction

Edge Contraction● To sew the small cracks,gaps,holes

Utilizing Vertex-Edge Contraction(description in this paper)

Background

Edge Contraction

Unifying two vertices lying on a common edge

Background

Vertex Contraction

Which is essentially a generalization of the edge contraction,the difference is that vertices not necessarily connected by an edge are contracted

Background

Vertex-Edge Contraction

The Operator of Vertex-edge Contraction

1. Project the vertex v orthogonally onto the edge e.

2.Insert a vertex v’ into the edge at the geometric position of the

projection.

3. Split the triangle t1 into two triangles t1’ = (v0,v’,v2) and t2 =

(v1,v2,v’).

4. Perform a vertex contraction to v and v’. The new position of

the vertex will be a convex combination of v and v’, we move

the new vertex Vnew into position lv+(1−l)v’.

Error Measure

• The most appropriate error measure is simply the Euclidean distance

Euclidean distance:

The straight line distance between two points. In a plane with p1 at (x1, y1) and p2 at (x2, y2), it is √((x1 - x2) ² + (y1 -

y2) ²).

Algorithm

1. degenerate faces without finite area

2. unwanted gaps and cracks between regions of the mesh resulting from erroneous scan data reconstruction or modeling and/or tessellation of analytical surfaces

3. holes in the model due to missing polygons

4. T-vertices lying on interior of an edge of a face

The mesh to be processed by our method will possibly include the following artifacts:

Algorithm

Preprocessing Phase

1. Reading the mesh

2. Identification of boundaries

3. Identification of corresponding vertex-vertex and vertex-edge pairs

4. Computation of error

5. Ordering according to error

Boundary Edges:

The edges having only one incident triangle

Boundary Vertices:

The vertices are simply incident to boundary edge

Algorithm

Decimation

1. Pop the boundary vertex or inner edge with minimal error from the priority queue. If the error e of the popped feature is larger than a prescribed error threshold e then STOP.

2. Perform the according operation, i.e. vertex-edge contraction for a boundary vertex, or an edge contraction for an inner edge.

Algorithm

Decimation

3. Encode the modification and store in the progressive data structure.

4. Recompute the error for vertices and edges influenced by the operation, accordingly maintain the priority queue. GOTO 1.

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